You should get most of the ranks through just playing the game and boosting the other trophies, you will most likely be around Rank 16 if you have been boosting most of the weapon kits and other multiplayer trophies.
There is a trick that you can do this by yourself if you are not good at the game.
Load up the map Cold War > Squad Rush > Hardcore Off. Get into the game and of course select the kit you want to level up too and run to objective A detonate it (you have to make sure you are on the Attacker side) then once the game opens up go to Objective B and detonate that too. Wait till the game loads the next round and puts you on the defence and allow it to save you need to do this to ensure the experience gained counts. Then quit out and repeat the process. You gain 1200 XP for everytime you do this and it takes about 3 to 4 minutes to do one round since most of the servers have no people playing these game types should get about 12,000 XP in about 35 minutes or so.
1 R001 Private 1 6,500 6,500
2 R002 Private 2 4,500 11,000
3 R003 Private 3 7,500 18,500
4 R004 Specialist 1 9,500 28,000
5 R005 Specialist 2 12,000 40,000
6 R006 Specialist 3 13,000 53,000
7 R007 Corporal 1 15,000 68,000
8 R008 Corporal 2 16,000 84,000
9 R009 Corporal 3 16,000 100,000
10 R010 Sergeant 1 20,000 120,000
11 R011 Sergeant 2 18,000 138,000
12 R012 Sergeant 3 20,000 158,000
13 R013 Staff Sergeant 1 21,000 179,000
14 R014 Staff Sergeant 2 21,000 200,000
15 R015 Staff Sergeant 3 24,000 224,000
16 R016 Master Sergeant 1 23,000 247,000
17 R017 Master Sergeant 2 25,000 272,000
18 R018 Master Sergeant 3 25,000 297,000
19 R019 First Sergeant 1 26,000 323,000
20 R020 First Sergeant 2 27,000 350,000
21 R021 First Sergeant 3 27,000 377,000
Warrant Officer Ranks
Rank No. Insignia Rank Points Required Total Points Required
22 R022 Warrant Officer 1 28,000 405,000